Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 2 / i / b

No normal flow at a rigid wall gives $\boxed{\varphi (\pm 1)=0}$. At a free surface $y=1+\eta$, the linearized kinematic boundary condition isso $(U-c)\eta =-\varphi$. The linearized tangential momentum equation gives the pressure amplitude $p=(U-c){\varphi }^{\prime }-U^{\prime }\varphi$. Constant surface pressure therefore requires